Are prayers invalidated if "Indeed, You are Praiseworthy and Glorious" is said only at the end of the second Tashahhud, and not after both the Salah and the Barakah?

Some scholars hold that invoking blessings upon the Prophet (peace be upon him) in the final tashahhud is a pillar [of the prayer], and that saying "Allahumma salli 'ala Muhammad" (O Allah, bless Muhammad) suffices, with anything beyond that being a sunnah. Therefore, not saying "Innaka Hamidun Majeed" (Indeed, You are Praiseworthy, Most Glorious) does not invalidate previous prayers, nor does it necessitate making them up.